Weighted Lattice Paths
Let $P_{a,b}$ denote a path in a $a\times b$ lattice grid with following properties: The path begins at $(0,0)$ and ends at $(a,b)$. The path consists only...
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John Rhodes
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August 16, 2023
Flexible Digit Sum
Given any positive integer $n$, we can construct a new integer by inserting plus signs between some of the digits of the base $B$ representation of $n$,...
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John Rhodes
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August 16, 2023
Subset Sums
Let $A_q(n)$ be the number of subsets, $B$, of the set $\{1, 2, …, q \cdot n\}$ that satisfy two conditions: 1) $B$ has exactly $n$ elements;...
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John Rhodes
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August 16, 2023
Numbers of the Form $a^2b^3$
Define $F(n)$ to be the number of integers $x≤n$ that can be written in the form $x=a^2b^3$, where $a$ and $b$ are integers not necessarily different and...
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John Rhodes
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August 16, 2023
Square Prime Factors II
For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of...
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John Rhodes
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August 16, 2023
Square Prime Factors
For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of...
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John Rhodes
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August 16, 2023
Constrained Permutations
Let $(p_1 p_2 \ldots p_k)$ denote the permutation of the set ${1, …, k}$ that maps $p_i\mapsto i$. Define the length of the permutation to be $k$;...
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John Rhodes
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August 16, 2023
Crossed Lines
Given a set, $L$, of unique lines, let $M(L)$ be the number of lines in the set and let $S(L)$ be the sum over every line of...